Activity Expected Time Variance

The expected time for the project is the sum of the preceding expected times because the path described is the critical path. Its value is 21 weeks. The variance of the end date is the sum of the variances of each activity, as shown

11.6 the uniform distribution 327

before, and is equal to 17/9, or 1.89. The standard deviation is its square root, or 1.37 weeks. If we now assume that the end-date distribution is normal, we can calculate the probability of the project exceeding 23 weeks, using the normal table and the fact that the mean (expected) value is 21 weeks. The time period between 23 and 21 weeks is 2 weeks, representing 1.46 sigma from the mean. From a table lookup at the 1.46 point, we obtain the area from the mean as 0.4279. The area to the right of that is therefore 0.5 -0.4279, or 0.0721. This corresponds to the probability that the project end date will exceed 23 weeks.

In the preceding example, it is demonstrated that the normal need not have a mean value of zero. If the distribution is shifted to the right by the value m, then it has a mean value equal to m and m is subtracted from the value of x in Equation (11.25). The mean value m and the standard deviation ct can be independently selected because it is a two-parameter distribution. A small standard deviation ''narrows'' the distribution and a large standard deviation broadens it.

The normal distribution is often used in analyzing communications systems to represent the noise distribution. As alluded to earlier, the noise power (N) is equivalent to the variance of the normal distribution, that is, N = ct 2. The normal distribution will be examined again in this chapter in the discussion of detecting a signal in noise. 